

\magnification = 2000 
\input amstex
\documentstyle{amsppt}
\input OurATOMacros
\input OurPlainGraphicsMacros



% File Name as ATO: Norm One Family

\Title  Norm One Family.
\medskip

\LF
Our family is  defined by the following implicit equation family:
$$
	\big(|x|^p + |y|^p + |z|^p \big)^{1/p} = 1, \hskip5mm 0 < p < \infty
$$
For $p \ge 1$ these surfaces bound the {\it convex} unit balls of the p-norms 
on $\BR^3$. For $p < 1$ these surfaces are {\it not} convex.
\Lf
$$\eqalign{
&p=1: \  \hbox{ Octahedron, }  \  \    |x|+|y|+|z|  \le 1. \cr
&p=2:  \  \hbox{ Round Sphere, }   x^2 + y^2 +z^2  \le 1.\cr
&p=\infty: \hbox{ Cube, } \hskip7.5mm \max(|x|,|y|,|z|)  \le 1.
}$$
\lf
The surfaces are all parametrized by polar coordinates, generalizing meridians
and lattitudes on the sphere (the Pascal values of $\sin(\pi), \cos(\pi/2)$ were not good 
enough for these pictures, they had to be redefined as $0$):
$$\eqalign{
&xp = \sin\theta\cos\phi,\ \ yp = \sin\theta\sin\phi, \ \ zp = \cos\theta, \cr
&x = \hbox{sign}(xp)|xp|^e, \ y = \hbox{sign}(yp)|yp|^e,   \cr
&z = \hbox{sign}(zp)|zp|^e,\cr
&2/p = e = 1 + \tan(ee),\ \  -\pi/4 <  ee < \pi/2 .
}$$
Look at the default $ee$-Morph in the View Menu.
\vfil\eject
\cl{Triangle Inequality and Convexity} \lf
A norm (or length) $\|.\|$ is a function on a real or complex vector space $\BV$ into 
$\BR_+$ that satisfies three axioms. For all $v,w\in\BV$ holds:
$$\leqalignno{
&  &\hbox{Definiteness:} \cr
&\|v\| \ge 0, \ \hbox{ and } \|v\| =0 \Rightarrow v=0.  &(A1) \cr
&  &\hbox{Homogeneity:} \cr
&  \|\lambda\cdot v\| = |\lambda|\cdot\|v\| \  \hbox{ for all } \lambda\in\BC. &(A2) \cr
&  &\hbox{Triangle Inequality:} \cr
&  \|v+w\| \le \| v \| + \|w\|.  &(A3) \cr
}$$
From these axioms follows the convexity of the unit ball $B:=\{ v; \ \| v \| \le 1 \} $. \lf
Proof: $v,w\in B,\ 0 \le t \le 1$ implies $(1-t)v + tw \in B$, since 
$$\eqalign{
&\|(1-t)v + tw \| \le \|(1-t)v\| + \|tw \| = \cr &\ \ (1-t)\|v\| + t\|w\| \le (1-t)+t =1.
}$$
The converse is also true: Every bounded closed convex set $B$ that is symmetric
($v\in B \Rightarrow -v\in B \ \hbox{resp. }\break\ni  e^{i\phi}v \in B$) 
and contains a neighborhood of $0\in\BV$ allows to define a norm as follows:
$$
  \|v\| := \sup\{1/|r| ;\  r\cdot v \in B \hbox{ for }  r\in\BR\}.
$$


\bye